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A few months ago I saw a TV crime drama that involved a plan to defraud bookmakers by nobbling the favourite for a race. The drama centred around other events and the basis for the betting fraud was never explained. What might have been going on?

Suppose that you have a race where there are published odds on the competitors of to 1, to 1, to 1, and so on for any number, , of runners. For example, if the odds are 5 to 4 then we express that as an of 5/4 to 1. Now if we lay bets on all of the runners in proportion to the odds so that we bet a fraction of the total stake money on the runner with odds of to 1 then we will always show a profit so long as the odds satisfy the inequality

and if is less than then our winnings will be at least equal to

Let's look at some examples. Suppose there are four runners and the odds for each are 6 to 1, 7 to 2, 2 to 1 and 8 to 1. Then we have and and

and so by betting our stake money with 1/7 on runner 1, 2/9 on runner 2, 1/3 on runner 3, and 1/9 on runner 4 we will win at least 51/63 of the money we staked (and of course we get our stake money back as well).

However, suppose that in the next race the odds on the four runners are 3 to 1, 7 to 1, 3 to 2 and 1 to 1 (ie evens). Now we see that we have

and there is no way that we can guarantee a positive return. Generally, we can see that if there is a large field of runners ( is big) there is is likely to be a better chance of being greater than 1. But large doesn't guarantee . Pick each of the odds by the formula and you can get and a healthy 30% return even when is infinite!

But let's return to the TV programme. How is the situation changed if we know ahead of the race that the favourite in our example will not be a contender because he has been doped?